Mathematical Problems in Engineering

Volume 2016, Article ID 7047126, 8 pages

http://dx.doi.org/10.1155/2016/7047126

## Series Solution for the Time-Fractional Coupled mKdV Equation Using the Homotopy Analysis Method

^{1}CONACYT-Centro Nacional de Investigación y Desarrollo Tecnológico, Tecnológico Nacional de México, Interior Internado Palmira S/N, Col. Palmira, 62490 Cuernavaca, MOR, Mexico^{2}Universidad Autónoma de la Ciudad de México, Prolongación San Isidro 151, Col. San Lorenzo Tezonco, Del. Iztapalapa, 09790 Ciudad de México, Mexico^{3}Centro Nacional de Investigación y Desarrollo Tecnológico, Tecnológico Nacional de México, Interior Internado Palmira S/N, Col. Palmira, 62490 Cuernavaca, MOR, Mexico

Received 22 June 2016; Accepted 8 September 2016

Academic Editor: Hang Xu

Copyright © 2016 J. F. Gómez-Aguilar et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We present new analytical approximated solutions for the space-time fractional nonlinear partial differential coupled mKdV equation. A homotopy analysis method is considered to obtain an infinite series solution. The effectiveness of this method is demonstrated by finding exact solutions of the fractional equation proposed, for the special case when the limit of the integral order of the time derivative is considered. The comparison shows a precise agreement between these solutions.

#### 1. Introduction

Many important phenomena occurring in different fields of physics, chemistry, biology, signal processing, system identification, control theory, and finance dynamics and many other problems in different areas are frequently modeled through fractional differential equations [1–14]. Several methods have been developed to study the solutions of nonlinear fractional partial differential equations (NFPDEs), for instance, the variational iteration method [15–17], Adomian decomposition method [18–20], fractional subequation method [21–23], the homotopy perturbation technique [24–27], the homotopy analysis method (HAM) [28, 29], and Laplace homotopy analysis method [30, 31]. However, for the nonlinear coupled equations with parameters derivative, especially fractional parameter derivative, not much work has been done [32].

A direct extension to the fractional-order case for the space-time fractional Hirota-Satsuma coupled mKdV equation takes the following form:where and are the Caputo derivatives [33], is a constant, and is the order of the fractional derivative.

The aim of this work is to investigate the approximated analytical solutions of (1). The obtained solutions will be compared with the solutions obtained previously in the literature [34–37], for the coupled equation (1). The outline of this work is as follows: in Section 2, we describe the basic tools of fractional calculus. Section 3 contains the application of the method to obtain the approximated solutions for the coupled mKdV fractional system. In Section 4, we compared our results with those reported in the literature [38]. Finally in Section 5 conclusions are given.

#### 2. Fractional Calculus

The Riemann-Liouville operator is defined as [39, 40]

The fractional derivative of in the Caputo sense is defined asfor , , , .

The fractional derivative of in the Caputo sense satisfies the following relations:

The Caputo operator is used here because the initial conditions for the fractional differential equations can be handled by using an analogy with the classical case (ordinary derivative).

#### 3. The HAM Applied to Coupled mKdV Equation

Consider the nonlinear space-time fractional coupled mKdV equationwith the initial conditions

The exact solutions obtained by applying the fractional subequation method are given by [41, 42]where

By means of HAM, we choose the linear operator [43]with and or . We define the nonlinear operator , , as

We construct the zero-order deformation equationwhere is a nonzero auxiliary parameter; when and , we have [44]

Now we obtain the th order deformation equationwhere

For the auxiliary function , we can take [44, 45]. Applying the Riemann-Liouville operator , the solution of the th order deformation equation (14) for becomes

Now if we substitute the initial condition (6), system (15) takes the following form:

Solving with Mathematica software, the approximate solutions of (5), considering [44], are

#### 4. Discussion

Next, we will compare the approximated analytical solution obtained in the above section with the exact analytical solution previously obtained in [41]. In this comparison, using Mathematica, setting in the solutions obtained by the HAM to the coupled mKdV system eq. (17), we can write for (classical case)which is the exact solution when , in the general solutions (7), that has been obtained previously, by applying the fractional subequation method [41]. Similar results have been obtained for the nonlinear fractional-order coupled mKdV equation when only time-fractional dependence has been considered in the HAM [36], but with integer order derivatives for space variables. However, in [36], they have compared their results with the exact analytical results obtained by Fan [38], for the classical case , that is,where in the classical solution of Fan the following dispersion relation was obtained [38]:

Figures 1 and 2 show the numerical solutions of the nonlinear coupled mKdV equations with , , at and . Figures 3 and 4 depict the exact solutions (7), with , , , , and .